A diode-pumped alkali laser (DPAL) is one of the most hopeful candidates to achieve high power performances. As the laser medium is in a gas-state, populations of energy-levels of a DPAL are strongly dependent on the vapor temperature. Thus, the temperature distribution directly determines the output characteristics of a DPAL. In this report, we developed a systematic model by combining the procedures of heat transfer and laser kinetics together to explore the radial temperature distribution in the transverse section of a cesium vapor cell. A cyclic iterative approach is adopted to calculate the population densities. The corresponding temperature distributions have been obtained for different beam waists and pump powers. The conclusion is thought to be useful for realizing a DPAL with high output power.
© 2014 Optical Society of America
Since W. F. Krupke in Lawrence Livermore National Laboratory (LLNL) invented the diode-pumped alkali laser (DPAL) at the beginning of the 21th century, such a new type of laser has been rapidly developed in the last decade [1–3].
In fact, DPALs combine the major advantages of solid-state lasers and gas-state lasers and obviate their main disadvantages at the same time [4–6]. Comparing to the traditional diode-pumped solid-state laser (DPSSL), DPALs have high Stokes Efficiency, good thermal performance, narrow linewidth, compact size, non-toxic system, etc [7–11]. Generally, the collisionally broadened cross sections of both the D1 and D2 lines are of the order of 10−13 cm−2 at 110 °C . Such values are much larger than the cross sections of stimulated radiation for the most conventional solid-state, fiber and gas lasers (e.g., about 106 times larger than a Nd:YAG laser) [13–16]. Thus, DPALs provide an outstanding potentiality for realization of high-powered laser systems. With so many marvelous merits, a DPAL becomes one of the most hopeful high-powered laser sources of next generation.
Unlike other types of lasers, the density of a gain medium inside a vapor cell is extremely sensitive to the ambient temperature [17–19]. When the pump power for a DPAL is small enough, heat generated in the vapor cell provides almost no effects on laser properties and is usually neglected. However, when a high-powered diode is used to pump the vapor cell, the laser features might become somewhat strange, and some physical characteristics, e.g. the transverse gain distribution, might be different from those for an ordinary solid-state laser.
Therefore, there is a necessity for DPAL technicians to investigate generated heat as well as the temperature distribution in the vapor cell for construction of a DPAL system with high beam quality. One of the greatest difficulties for examination of thermal characteristics of a DAPL cell is that the population status exhibits an inhomogeneous distribution due to the temperature gradient inside an alkali cell. Simultaneous evaluations of both the population and the temperature distributions cannot be simply completed only with the analyses of thermal conductivity or kinetic calculations. Until now, two teams have carried out studies on the temperature distribution inside an alkali vapor cell. One is the team headed by B. L. Pan, whose interests concentrate on the heat transfer and the optical path difference (OPD) under different pump powers [20, 21]. They calculated the temperature distribution by using an assumed absorption coefficient, and the lasing process was not taken into account in their scheme. The other is B. D. Barmashenko's team and their calculation model was only based on the kinetic evaluation while the temperature distribution was not discussed [22–25]. In their model, the temperature is assumed as a constant inside the lasing region.
To deduce an accurate temperature distribution inside a vapor cell, it is essential to create an analytic system by considering both the laser kinetics and the heat transfer. In our report, the heat generating dynamics and heat transfer are investigated by two interrelated theoretical procedures. The results reveal that the radial temperature gradient inside the vapor cell cannot be simply ignored for a real DPAL. To the best of our knowledge, there have not been any similar reports on this topic so far.
2. Theory and method
For an end-pumped configuration, the optical axis of a pump laser diode coincides with that of a DPAL. As shown in Fig. 1, we divide a cylindrical vapor cell into many cylindrical annuli whose axes are same. Every cylindrical annulus is thought as a heat source. These coaxial cylindrical annuli will be used in the segmental accumulation procedure as introduced next.
During the evaluation, we made the following assumptions:
- (1) The diameter of the DPAL beam is approximately treated to be unchanged along the optical axis;
- (2) The transverse pump distribution holds out a Gaussian intensity profile and keeps unchanged along the optical axis;
- (3) The temperature of every cylindrical annulus is a constant along the optical axis;
- (4) The effects of both end-windows of the enclosed vapor cell are ignored.
Actually, the variations in diameter of both the DPAL and the pump beams are very tiny inside a 2.5cm-long cell. For example, the variation of the beam size is only about 2.2% for a pump beam with the waist size of 150 μm. In this research, the temperature distribution along the axial direction is ignored with purpose of the algorithm simplification. We will undertake the evaluation of an enclosed cell to deduce the realistic 3-dimensional temperature distribution, in which the effects of both end-windows are taken into account in the evaluation regime.
Heat transfer in a gas-state medium exhibits three types: heat radiation, convection, and conduction. Since the thickness of every cylindrical annulus is very small, heat transfer between the annulus gap is mainly dominated by heat conduction and the effects of natural convection is neglected in such a narrow gap [26, 27]. Additionally, heat radiation is so small that it can also be ignored here.
2.1. Analyses of laser kinetics
Generally, a DPAL is thought to be a typical three-level laser source with a tiny quantum defect. As shown in Fig. 2, the stimulated radiation transition n2P1/2→n2S1/2 is called D1 line and the stimulated absorption transition (pump transition) n 2S1/2→ n 2P3/2 is called D2 line, where n = 4, 5 and 6 for K, Rb and Cs, respectively . The D2 line can be collisionally broadened to achieve spectrally homogeneous transition by using a buffer gas such as helium. In the presence of helium buffer gas, the D2 pump transition line-shape changes from Gaussian to Lorentzian. Thus, pump energy absorbed in the spectral wings of the pump transition can dramatically enhance the laser gain by comparing the case of no helium buffer gas. The effectively collision-broadened linewidth is generally at least ten times the Doppler linewidth. The relaxation rate of the fine-structure can be enhanced by adding some alkanes with small hydrocarbon molecules [18, 19, 28]. In the theoretical simulation of this report, we choose cesium as the laser gain medium, and helium and ethane as buffer gases.
As shown in Fig. 3, we select an arbitrarily cylindrical annulus (jth) among the segments. The outside radius rj and the inner radius rj + 1 of this cylindrical annulus can be simply expressed by12]12]29]12]12]12]
In addition, the number density of every energy-level must satisfy the following two equations for a steady-state laser emission :22]30]
Next, we calculate the volume density of generated heat of the jth cylindrical annulus by using the following formula :
Thus, the generated heat of the jth cylindrical annulus can be obtained by the following calculation:
2.2. Theoretical analyses of heat transfer
2.2.1. Calculation of a Transverse Section Except the Central Core
Generally, the differential equation of thermal conductivity in the cylindrical coordinate system is given by 24]32]
Next, we get the following formula by undertaking the integral calculation on both sides of Eq. (21):31]Eqs. (24) and (26) into Eq. (25), can be calculated by the following formula:Eq. (18) of Subsection 2.1. Then, we make a further integral calculation on both sides of Eq. (24) and the temperature distribution inside the jth cylindrical annulus can be given byEq. (28):Eq. (28), one can obtain the temperature distribution in the transverse section of the jth cylindrical annulus. The temperature of the inner side of the jth cylindrical annulus, , can also be deduced and is then used as the board condition in the calculation of the (j + 1)th cylindrical annulus. By employing a circulatory calculation, we can therefore obtain T2, T3, …,TN.
2.2.2. Calculation of Temperature of the Central Core
The exterior radius rN of the central cylinder (core) can be simply expressed by
The temperature distribution inside the central core can be given by
can be solved by substituting r = rN, T = TN into Eq. (32),
The central temperature of the vapor cell, TN + 1, can be obtained by substituting into Eq. (32). By combing the results of Eqs. (28) and (32) together, it is possible to acquire the whole temperature distribution at the cross-section of the vapor cell.
2.3. Calculation of radial temperature distribution
The flowchart for evaluation of the temperature distribution is diagramed in Fig. 4. First, we assume that the total heat transferred out from a vapor cell is PThermal . By considering a fact that the heat delivered from the first cylindrical annulus to the cell wall, Φ1, is equal to the total heat release, the following relationship is tenable:Eq. (28).
The temperature of the inner side of the first cylindrical annulus, T2, can be then evaluated and is utilized as the initial conditions in calculating the temperature distribution of the second cylindrical annulus. As depictured in Fig. 5, the heat transferred from the second cylindrical annulus to the first one, Φ2, is thus calculated by
Therefore, through a circulatory calculation of Q1, Q2, …, QN-1, we can obtain heat transferred from the jth cylindrical annulus to the (j-1)th one as expressed by
Next, we judge whether is equal to the given value of PThermal or not. If the answer is “no”, the evaluation will be repeated by using the next value of PThermal until the following equation is satisfied (see Fig. 6):Fig. 6 represents the heat generated from all cylindrical annulii or the heat transferred outside from the first cylindrical annulus. By using the correct value PThermal, the temperature distribution of the vapor cell can be obtained in the transverse section.
3. Results and discussions
3.1. Population distributions
3.1.1. Different waists of pump beams
We first analyze the population distributions inside a cesium vapor cell for different waists of pump beams. By using the approach introduced in Section 2, we calculate the population density distributions inside the cell as illustrated in Fig. 7. During the evaluation, the pump power is fixed to 10 W and the other parameters are listed in Table 1. The waist radii are assumed as 150 μm, 300 μm, 500 μm, and 700 μm corresponding to (a), (b), (c), and (d) in Fig. 7, respectively. It is observed that the total density of the cesium vapor n0 increases with the radial position r. Such a phenomenon is due to the fact that the temperature at the central area is higher than that near the wall of the vapor cell. Some significant variation of the population densities n1 and n2 can be seen in the figure. The inflection points in the legend give rise to discontinuity of the first derivative or “angles” on the curves for n1 and n2. Such inflection points located in the lasing boundary line. In the lasing region, n2 is always larger than n1 because of population inversion. We also find that, the bigger the spot size of a pump beam is, the lower n3 becomes. It means that more electrons will be stimulated into the 6 2P3/2 level under a higher pump density. However, it also leads to a population accumulation at the top energy-level. One can realize that the higher pump density brings about a relative weak relaxing capability by comparing n3 and n2 in (a), (b), (c), and (d) of Fig. 7.
3.1.2. Different pump power
Next, we discuss the population distributions inside a cesium vapor cell for different pump power when the beam waist radius is fixed to 500 μm. By using the method mentioned above, we get the results when the pump power is 1 W, 20 W and 50 W, respectively. As diagramed in Fig. 8, the total number population n0 decreases with the pump power. The reason is that the central temperature increases with the pump power and the total population density generally exhibits a degressive tendency with the temperature rising by referring Eq. (17). In the calculation, lasing output corresponding to the population distribution in Fig. 8(a) cannot be achieved because the threshold condition is unsatisfied.
3.2. Temperature distributions
By use of the approach introduced in Section 2, the radial temperature distributions are obtained. As shown in Fig. 9, the temperature at the cross section exhibits a distinct gradient and achieves the maximum values at the central axis for every case. Note that the waist radius of the pump beam in Fig. 9(a) is 150, 300, 500 and 700 μm, respectively, when the pump power is set to 10 W. We can observe that the curve for ωp = 500 μm is almost located at the lowest position at the diagram. It means that a big pump density will not always cause a high heat generation. To make the expression clear, we also produced a 3-Dmensional drawing for ωp = 500 μm as depictured in Fig. 9(b). In Fig. 10, the pump powers is set to 1, 10, 100, and 500 W, respectively, when the waist radius of the pump beam is 500 μm. It is obvious that the temperature gradient increases rapidly with the pump power. Such tendencies can be explained by a fact that the thermal conductivity of a gas-state medium is so small that the generated heat cannot be transferred outside efficiently.
In Fig. 10, it is easily found that a higher pump power will cause a more distinct temperature gradient. According to our previous study  and the experimental results in Ref. 34, there should be an optimum temperature for achieving the highest laser output. Therefore, investigating how the conversion efficiency changes with the pump power becomes a valuable business. In Fig. 11(a), both the absorbed power and the generated heat inside a cesium vapor cell are given as a function of the pump power. Both quantities monotonically increase with the pump level and only less than 5% of the absorption converts to heat. In Fig. 11(b), the optical-optical conversion efficiency first raises as the pump power increases, and then it gradually slows down. This is because that n3 becomes higher in this case, and decrease in the population densities n1 and n2 will lead to a relatively low conversion efficiency.
It is interesting to find that the peak position of the optical-optical conversion efficiency corresponds to a relatively low output. The optimum pump power is around 30 W for the optical-optical conversion efficiency but the laser output is only 13.4 W. The results are different from some traditional solid-state lasers. Thus, the output characteristics of a DPAL are not only dependent on the energy levels, but also determined by the thermal features.
Until now, the temperature inside the vapor cell has been considered as a constant in most of the literatures of DPALs. Actually, it is impossible completely eliminate the temperature gradient of a heated gas-state medium because of its small thermal conductivity. Since the saturated density of the alkali vapor is strongly dependent on the temperature, the kinetics process should be taken into account during evaluating the thermal characteristics. In this study, we developed a scheme by using both the heat transfer and the laser kinetics to analyze the temperature distribution inside an enclosed vapor cell. In our theoretical model, a cell is segregated to many coaxial cylindrical annuli. The population density and temperature in every cylindrical annulus have been finally calculated. It is seen that the temperature in the center of a vapor cell is higher than the other places at the cross section. The temperature gradient becomes serious with increase of the pump intensity. However, unlike some of the solid-state lasers, a stronger pump power does not always result in a higher optical-optical conversion efficiency for a DPAL. To solve such a problem, one of the effective methods to achieve a high-powered DPAL might be adopting a flowing-gas system. The thermal effects should be dramatically decreased by using such a dynamic procedure. The mathematical model will be much more complicated than that used in this report. In fact, the theoretical model in this paper can also be applied to the other two types of DPALs if the D1 and D2 radiative lifetimes, the pump absorption cross section, the collisionally-broadened cross section, and the thermally averaged relative velocities are changed to the new values corresponding to rubidium and potassium.
Additionally, the temperature distribution will inevitably bring about some changes in refractive index with temperature dn/dT. Decreasing the thermally-induced wave front distortion and thermal lensing effects should be important in realization of a DPAL with high beam quality. We will also introduce our results on this issue afterwards.
We are very grateful to Prof. Salman Rosenwaks and Dr. Boris D. Barmashenko at Ben-Gurion University of the Negev of Israel for their valuable helps in calculation of saturated alkali number densities inside a static alkali vapor cell.
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